Find Dⁿ{x²y₂ + x y₁ + y}

This post gives the standard closed-form result for the nth derivative of x²y'' + x y' + y using Leibniz’s theorem (nth derivative of a product).

Meaning of symbols

• Dⁿ means the nth derivative with respect to x.
• y₁ = dy/dx, y₂ = d²y/dx², and in general y_k = d^ky/dx^k.

Leibniz theorem (product rule for nth derivative)

If u(x) and v(x) are differentiable n times, then:

dⁿ/dxⁿ(u·v) = Σ_r=0ⁿ (nCr) · u^(n−r) · v^(r)

(This is the standard statement used in successive differentiation problems.)

Source: Leibniz theorem statement and formula. [web:22]

Final answer

Using Leibniz’s theorem on each term and simplifying, the nth derivative is:

Dⁿ{x²y₂ + x y₁ + y} = x²y_n+2 + (2n+1)xy_n+1 + n²y_n.

Leibniz theorem is used to expand nth derivatives of products. [web:22]